### Abstract

It is common to classify satisfiability problems by their time complexity. We consider another complexity measure, namely the length of certificates (witnesses). Our results show that there is a similarity between these two types of complexity if we deal with certificates verifiable in subexponential time. In particular, the well-known result by Impagliazzo and Paturi [IP01] on the dependence of the time complexity of k-SAT on k has its counterpart for the certificate complexity: we show that, assuming the exponential time hypothesis (ETH), the certificate complexity of k-SAT increases infinitely often as k grows. Another example of time-complexity results that can be translated into the certificate-complexity setting is the results of [CIP06] on the relationship between the complexity of k-SAT and the complexity of SAT restricted to formulas of constant clause density. We also consider the certificate complexity of CircuitSAT and observe that if CircuitSAT has subexponential-time verifiable certificates of length cn, where c < 1 is a constant and n is the number of inputs, then an unlikely collapse happens (in particular, ETH fails).

Original language | English |
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Title of host publication | Theory and Application of Satisfiability Testing - 14th International Conference, SAT 2011, Proceedings |

Pages | 19-32 |

Number of pages | 14 |

DOIs | |

Publication status | Published - 27 Jun 2011 |

Event | 14th International Conference on Theory and Applications of Satisfiability Testing, SAT 2011 - Ann Arbor, MI Duration: 19 Jun 2011 → 22 Jun 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6695 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 14th International Conference on Theory and Applications of Satisfiability Testing, SAT 2011 |
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Country | United States |

City | Ann Arbor, MI |

Period | 19/06/11 → 22/06/11 |

### Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theory and Application of Satisfiability Testing - 14th International Conference, SAT 2011, Proceedings*(pp. 19-32). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6695 LNCS). https://doi.org/10.1007/978-3-642-21581-0_4